Sensitivity for four-body tau-lepton decays at Belle and Belle II experiments

We study the expected sensitivity at Belle and Belle II for four-body $\tau^\mp \to X^\pm l^\mp l^\mp \nu_{\tau}$ decays where $l=e$ or $\mu$ and $X=\pi$, $K$, $\rho$ and $K^*$ mesons. These decay processes violate the total lepton number ($|\Delta L|=2$ ) and they can be induced by the exchange of Majorana neutrinos. In particular, we consider lifetimes in the accessible ranges of $\tau_N$ = 5, 100 ps and extract the limits on $|V_{\ell N}|^2$ without any additional assumption on the relative size of the mixing matrix elements. For an integrated luminosity collected of 1 ab$^{-1}$ at Belle, we found significant sensitivity on branching fractions of the order BR($\tau^\mp \to X^\pm l^\mp l^\mp \nu_{\tau}$) $\sim 10^{-8}$. For an integrated luminosity expected of 50 ab$^{-1}$ and intermediate luminosity of 10 ab$^{-1}$ at the Belle II, we found significant sensitivity on branching fractions of the order BR($\tau^\mp \to X^\pm l^\mp l^\mp \nu_{\tau}$) $\sim 10^{-9}-10^{-8}$. We use these sensitivities to set limits for the exclusion regions on the parameter space $(m_N, |V_{\ell N}|^2)$ associated with the heavy neutrino; such that for a $|V_{\ell N}|^2 \sim \mathcal{O}(10^{-5})$ at $\tau_N = 100$ ps, we find the bounds as $0.140 < m_N < 1.776$ GeV for $\tau^- \to X^+ e^- e^- \nu_\tau$ and $0.245 < m_N < 1.671 $ GeV for $\tau^- \to X^+ \mu^- \mu^- \nu_\tau$.